Convexity of the Bergman Kernels on Convex Domains

TL;DR

This paper proves the convexity of the logarithm of weighted Bergman kernels on convex domains in complex space and establishes related inequalities, advancing understanding of their geometric properties.

Contribution

It introduces new convexity results for weighted Bergman kernels on convex domains and derives a Brunn-Minkowski type inequality, with conditions for strict convexity.

Findings

Logarithm of weighted Bergman kernel is convex on convex domains.

A Brunn-Minkowski type inequality for the Bergman kernel is established.

Conditions for strict convexity of the kernel are characterized.

Abstract

Let $\Omega$ be a convex domain in $\mathbb{C}^n$ and $\varphi$ a convex function on $\Omega$. We prove that $\log{K_{\Omega,\varphi}(z)}$ is a convex function (might be identically $-\infty$) on $\Omega$, where $K_{\Omega,\varphi}$ is the weighted Bergman kernel. When $\varphi\equiv0$, we prove a Brunn-Minkowski type inequality, which further implies that $K_\Omega(z)^{-\frac{1}{2n}}$ is a convex function if $\Omega$ is convex. Some necessary and sufficient conditions for strictly convexity are also obtained.